I've used two techniques in pulling stumps with a backhoe. The first move for either technique is to cut the stump as close to the ground as possible. This allows better access to the far side of the stump. The technique choice depends on the general root structure.
For stumps with a shallow spreading root structure (Pine, sycamore etc):
Position the tractor so the stump is in the middle of the backhoe reach. Start scraping the soil at a distance of three to five stump diameters from the stump. The roots are smaller and easier to break further away from the stump. Once you have broken a root it is easier to break it again closer to the stump. You may have to take material out below the larger roots so you can get the bucket under them to break them with lifting force.
Repeat this around the stump. Plan ahead. put the spoil pile where it will be as much out of the way as possible. Don't hesitate to move the tractor if you think a different location would give you better purchase on the roots.
Sometimes, especially if the tree was growing on the edge of a down slope, the surface roots will go down, following the soil contours. These are tough to deal with because the slope is not a good place to put a backnoe. Of the fifty stumps I've pulled recently, 36 of them were on the edge of a downslope. I used the cutter edge between the teeth of the bucket to scrap the roots down, weakening them until I could break them with lifting force. This isn't easy when you are working at full reach, but that is what you end up having to do.
Digging under the stumps so that you can pull the stumps towards you will help in some cases. Grabbing roots that have broken free with the bucket and swinging the backnoe back and forth can weaken the remaining roots. The more degrees of freedom you can give the stump the easier it is to break the remaining root structure.
Trees with tap roots (many nut trees etc):
Get closer to the stump, you're going to be digging deeper. Dig down as close to the stump as you can. Do both sides and then move the backhoe so that you can dig as deep on the other two sides. Dig under the stump as far as you can. Try breaking the tap root as far down in the hole as you can reach. You may end up doing a lot of scraping before it breaks.
That about covers the _normal_ situations. Things can get pretty weird underground.
I've encountered boulders that were bigger than the stumps that had to be removed before I could free the stump. I found a 7 foot long granite billet with tree roots wrapped around it (that was a major pain).
I cut a cluster of maple trees down and found that they were suckers from the crown of s stump that had been knocked over and buried 3 feet down over 30 years ago. The buried stump was bigger around than the largest of the trunks in the cluster. I ended up digging down about 8 feet and the opening on the hole was about 12 feet before I ended up climbing into the hole with a chainsaw and cutting the buried stump to free the cluster's stump.
Another strange case was a yellow pine stump. When I started digging I found that there were no surface roots. The tree had been standing when the area was backfilled and the true crown was six feet underground. I dug the hole as you would for a tap root tree and went in with the chainsaw again. I took out the trunk section and covered the stump.
BTW, you get one cut with the chainsaw. No matter how clean you try to get the root or underground trunk section there will be enough grit in the path of the chain to dull it.
I hope this helps.
Matthew